Leslie Matrix

λ (lambda) is the finite rate of increase: the factor by which the population multiplies per time step (e.g., year). λ = N(t+1) / N(t). r is the instantaneous rate of increase: ln(λ). Both describe growth: λ > 1 means exponential growth, λ = 1 means stable, λ < 1 means decline. r > 0 means growth, r = 0 means stable, r < 0 means decline. They are equivalent: r = ln(λ).

Elasticity of λ to a vital rate is the proportional change in λ divided by the proportional change in the vital rate: e_ij = (a_ij / λ) * (∂λ / ∂a_ij), where a_ij is the matrix element. It answers: if I increase survival of age class x by 10%, how much does λ increase? Elasticities sum to 1 and reveal which life stages most influence growth. Focus conservation on the life stages with highest elasticity.

The stable age structure is the proportion of the population in each age class as time goes to infinity (the dominant right eigenvector). It emerges regardless of initial age structure if vital rates are constant. It matters because it describes the population's long-term composition: a population with an aging stable age structure (few young, many old) has lower growth potential than one with a young structure. Compare observed age structure to the stable structure to assess whether the population is currently above or below stable growth.

Deterministic Leslie matrices assume constant vital rates. Stochastic Leslie matrices randomly vary vital rates each time step to represent environmental fluctuations. Stochastic projection produces distributions of possible outcomes rather than single trajectories. If environmental variation is high, stochastic models predict lower average growth and higher extinction risk than deterministic models. Use stochastic projections for conservation decision-making.

Age zero is tricky because the Leslie matrix assumes vital rates are averaged over one time unit. Fertility rates in the first row apply to age 0 females (they give birth at the end of the interval). Survival on the subdiagonal is the probability of surviving from age i to age i+1. Document your convention clearly. Some software distinguishes age-of-birth from age-in-time to clarify the timing of vital rates.